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Unformatted text preview: e delivered at the same time, in one ‘shot’, with the
146 treatment organized as a sequence of ‘shots’.) We let bj denote the level of beam j , for j = 1, . . . , n.
These must satisfy 0 ≤ bj ≤ B max , where B max is the maximum possible beam level. The exposure
area is divided into m voxels, labeled i = 1, . . . , m. The dose di delivered to voxel i is linear in
m
the beam levels, i.e., di = n=1 Aij bj . Here A ∈ R+ ×n is a (known) matrix that characterizes the
j
beam patterns. We now describe a simple radiation treatment planning problem.
A (known) subset of the voxels, T ⊂ {1, . . . , m}, corresponds to the tumor or target region. We
require that a minimum radiation dose Dtarget be administered to each tumor voxel, i.e., di ≥ Dtarget
for i ∈ T . For all other voxels, we would like to have di ≤ Dother , where Dother is a desired maximum
dose for non-target voxels. This is generally not feasible, so instead we settle for minimizing the
penalty
E=
((di − Dother )+ )2 ,
i∈T where (·)+ den...
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